11ProbHw8sol

# 11ProbHw8sol - E Y = E X Â Y/X = E X E Y/X = l 2 1 2 = l 4...

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IEOR 3658 Assignment #8 Solutions Probability November 13, 2011 Prof. Mariana Olvera-Cravioto Page 1 of 2 Assignment #8 Solutions 1. (a) Since Andrew put \$ y in the meter, he has paid for s = 2 y hours of parking. If T is the time he spends at the dentist, then he will only get a ticket if T > s , in which case the number of hours between the time the meter runs out and the time Andrew comes out of the dentist is X = T - s . Therefore, P ( A ) = Z s P ( A | T = t ) f T ( t ) dt = Z s P ( A | X = t - s ) f T ( t ) dt = Z s (1 - e - 0 . 5( t - s ) ) e - t dt = Z s e - t dt - 1 1 . 5 e 0 . 5 s Z s (1 . 5) e - 1 . 5 t dt = e - s - 2 3 e 0 . 5 s e - 1 . 5 s = 1 3 e - s = 1 3 e - 2 y (b) Andrew’s expected expense when he puts \$ y in the meter is c ( y ) = y + 25 P ( A ) = y + 25 3 e - 2 y From where we get that c 0 ( y ) = 1 - 50 3 e - 2 y and c 00 ( y ) = 100 3 e - 2 y > 0 for all y. Hence, c ( y ) has a unique minimum at y = 1 2 log ± 50 3 ² 1 . 40 so he should put \$1.40 in the meter, which pays for 2 hours 48 min. 2. (a) f X,Y ( x, y ) = 1 xl , 0 x l , 0 y x (b) f Y ( y ) = R l y 1 xl dx = 1 l ln ( l y ), 0 y l (c) E [ Y ] = R l 0 1 xl y dx = l 4 (d) It is easy to see that R.V. Y/X is uniformly distributed on [0,1] and independent of X. Therefore,

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2 IEOR 3658, Assignment #8 Solutions
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Unformatted text preview: E [ Y ] = E [ X Â· Y/X ] = E [ X ] E [ Y/X ] = l 2 1 2 = l 4 3. (a) Z 1 Z 1 1-y cxy dxdy = c Â· 1 2 Z 1 y [1-(1-y ) 2 ] dy = c 2 Z 1 2 y 2-y 3 dy = c 2 (2 Â· 1 3-1 4 ) = 5 c 24 = 1 Therefore, c = 24 5 (b) Z 3 4 1 2 Z 1 2 1-y cxy dxdy = 53 1280 (c) Z 1 1-x cxy dy = cx 1 2 (2 x-x 2 ) = 12 5 (2 x 2-x 3 ) 4. p X ( k ) = P ( X = k ) = Z âˆž P ( X = k | Î» = m ) f Î» ( m ) dm = Z âˆž m k k ! e-m e-m dm = Z âˆž m k k ! e-2 m dm = 1 2 k +1 k = 0 , 1 , ... 5. (a) P ( X < Y ) = R 1 R 1 x f ( x, y ) dydx (b) P ( X + Y â‰¤ 1) = R 1 R 1-x f ( x, y ) dydx (c) Since X and Y are continuous random variables, it is the same as that in (a). (d) P ( XY â‰¤ 1 / 2) = R 1 2 R 1 f ( x, y ) dydx + R 1 1 2 R 1 2 x f ( x, y ) dydx...
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## This note was uploaded on 12/11/2011 for the course IEOR 3658 taught by Professor Olvera during the Fall '08 term at Columbia.

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11ProbHw8sol - E Y = E X Â Y/X = E X E Y/X = l 2 1 2 = l 4...

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